The Higgs boson has mass 125.09±0.21 GeV. You might see a statement that a 95% confidence interval for the mass is [124.88,125.30], and figure that physicists are 95% sure that the mass is within that interval. Or that 95% of the observations were within that interval.

Nope. It has some more roundabout definition. It does not directly give you confidence that the mass is within the interval.

Why prostitute my mind? I should start charging. By the way, The Cult of Statistical Significance by Stephen Ziliak & Deirdre McCloskey is a great book on the problem with Fischerian vs Gossetian statistics.